Sadly, the fastest reaction to destroy the Cryogunner shield is indeed Pyro. So, if you don't have Pyro at hand, either you find a way to be creative or you have to resort to slowly abuse reactions that will do thick damage past the shield.
One option that comes to mind is to use some external source of Pyro damage. If for example you can have another enemy ignite some grass, then swirl could do wonders - but it does require some other enemy that uses Pyro. Another option is to use attacks that can partially bypass shields (Xiao Bane of All Evil comes to mind, Jean Gale Blaze can also slowly work), but those are also not always available. Other than that... there aren't many other options right now.
Some players currently believe that the missing Dendro element will be tied to poison/damage over time and that could mean it will bypass shields somehow (and considering the new mechanics that bypass player shields in Abyss that could make sense) but until then, your best option is sadly to bail out and return with a Pyro character if you can.
To really understand how shields work, you can refer to the wiki here.
Over-simplified version follows.
There are three types of shields:
- "Armor" type shield: gives increased defense but does not nullify damage. Example: Lawachurls
- "Force" type shield: gives full damage immunity until broken. Example: Abyss Mage.
- "Directional" shield: those are the actual shields some mobs hold, they only defend from a specific direction.
The third type can be safely ignored for this argument
A shield can be reduced either by damage or elemental reactions. Both scale with a factor that is dependent on the shield element and mob type.
Fatui Skirmisher shields are considered "Armor" shields (with some further differences for the Anemoboxer) - this is why you can cause damage to their hp even when the shield is on. In most cases Armor shields have a damage ratio of 0, meaning that raw damage does not decrease the shield gauge.
For this reason damage won't reduce a Cryogunner shield gauge, leaving reactions as the only way to counter the shield.
With that said, let's look at the standard elemental damage coefficient table.
Again, skipping the actual formula, this should easily show that to destroy a Pyro shield Hydro is the best option, followed by Electro (again, ignoring other factors like application speed and so on).
Problem is that while basically every enemy shield uses this table, there are exceptions. And Skirmisher are one of those.
And here is the issue I mentioned. Skirmisher reaction coefficient is 2 for a specific element and "almost zero" for every other option. So, while for regular shields the max difference is a 4:1 ratio (Pyro VS Anemo on a Cryo shield)... for the Fatui the ratio can be as bad as 40:1 (Pyro vs anything else on the same Cryo shield) - ten times as bad. This basically means that with the sole exception of a single, working element, most reaction are basically "useless" at removing the shield.